While those exact examples may not work in all cases. However, I do get the idea.

In the case of the sequence, the subscript is approaching infinity.
Therefore, we take the maximum N to insure all conditions hold.

The case of continuity is a bit more abstract. In choosing the we are creating a bound on x, we want to be able to say that .
Knowing that means .
This means we can construct bounds on .
So we want to be at most 1 (not more).